In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u + λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k(Ω) in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞).
In this paper, the fundamental solution of rotating generalized Stokes problem in R^3 is established. To obtain it, some fundamental solutions of other problems also are established, such as generalized Laplace problem, generalized Stokes problem and rotating Stokes problem.
The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided.
In this paper, we consider a linearly elastic shell, I.e. A three-dimensional linearly elastic body with a small thickness denoted by 2ε, which is clamped along its part of the lateral boundary and subjected to the regular loads. In the linear case, one can use the two-dimensional models of Ciarlet or Koiter to calculate the displacement for the shell. Some error estimates between the approximate solution of these models and the three-dimensional displacement vector field of a flexural or membrane shell have been obtained. Here we give a new model for a linear and nonlinear shell, prove that there exists a unique solution U of the two-dimensional variational problem and construct a three-dimensional approximate solutions UKT(x,ξ) in terms of U:{ UKT(x,ξ):=U(x)+П1Uξ+П2Uξ2,П1U=-aαβ*▽βU3→eα-λ0γ0(U)→n,П2U=(1/λ+μ)*▽β(aαβλσ+γλσ(U))-bαβ*▽βU3)→eα+1/2λ0(ρKT0(U)-(1+λ0)Hγ0(U)-2β0(U))→n.We also provide the error estimates between our model and the three-dimensional displacement vector field:‖u-UKT‖1,(Ω)≤Cεr, r=3/2, an elliptic membrane, r=1/2, a general membrane,where C is a constant dependent only upon the data ‖u‖3,Ω, ‖UKT‖3,Ω, →θ.